Cauchy-Binet for Pseudo-Determinants

@article{Knill2013CauchyBinetFP,
  title={Cauchy-Binet for Pseudo-Determinants},
  author={Oliver Knill},
  journal={arXiv: Rings and Algebras},
  year={2013}
}
  • O. Knill
  • Published 1 June 2013
  • Mathematics
  • arXiv: Rings and Algebras

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References

SHOWING 1-10 OF 61 REFERENCES

On the Coefficients of the Characteristic Polynomial

It is a well-known fact that the first and last non-trivial coefficients of the characteristic polynomial of a linear operator are respectively its trace and its determinant. This work shows how to

The McKean-Singer Formula in Graph Theory

The McKean-Singer formula chi(G) = str(exp(-t L)) which holds for any complex time t, and is proved to allow to find explicit pairs of non-isometric graphs which have isospectral Dirac operators.

Matrix-Forest Theorems

A graph-theoretic interpretation is provided for the adjugate to the Laplacian characteristic matrix for weighted multigraphs and the analogous theorems for (multi)digraphs are established.

The zeta function for circular graphs

It is proved that the roots of zeta(n,s) converge for n to infinity to the line Re(s)=1/2 in the sense that for every compact subset K in the complement of this line, and large enough n, no root of the zeta function zeta(-s) is in K.

Selfsimilarity in the Birkhoff sum of the cotangent function

We prove that the Birkhoff sum S(n)/n = (1/n) sum_(k=1)^(n-1) g(k A) with g(x) = cot(Pi x) and golden ratio A converges in the sense that the sequence of functions s(x) = S([ x q(2n)])/q(2n) with

On the Dimension and Euler characteristic of random graphs

It is shown here that the average dimension E[dim] is a computable polynomial of degree n(n-1)/2 in p and the explicit formulas for the signature polynomials f and g allow experimentally to explore limiting laws for the dimension of large graphs.

Minors of the Moore-Penrose inverse

The Theory of Matrices

Volume 2: XI. Complex symmetric, skew-symmetric, and orthogonal matrices: 1. Some formulas for complex orthogonal and unitary matrices 2. Polar decomposition of a complex matrix 3. The normal form of

Self-similarity and growth in Birkhoff sums for the golden rotation

We study Birkhoff sums with at the golden mean rotation number with continued fraction pn/qn. The summation of such quantities with logarithmic singularity is motivated by critical KAM phenomena

An Introduction to Linear Algebra

Class notes on vectors, linear combination, basis, span. 1 Vectors Vectors on the plane are ordered pairs of real numbers (a, b) such as (0, 1), (1, 0), (1, 2), (−1, 1). The plane is denoted by R,
...